Question 60
The sequence $a_1, a_2, \dots$ satisfies the equality $\sum_{i=1}^{n} a_{\lfloor \frac{n}{i} \rfloor} = n^{10}$ for each positive integer $n$. Given an arbitrary positive integer $c$, prove that $\frac{c^{a_n} - c^{a_{n-1}}}{n}$ is an integer for each $n \gt 1$.